1,317 research outputs found
Fisher Lecture: Dimension Reduction in Regression
Beginning with a discussion of R. A. Fisher's early written remarks that
relate to dimension reduction, this article revisits principal components as a
reductive method in regression, develops several model-based extensions and
ends with descriptions of general approaches to model-based and model-free
dimension reduction in regression. It is argued that the role for principal
components and related methodology may be broader than previously seen and that
the common practice of conditioning on observed values of the predictors may
unnecessarily limit the choice of regression methodology.Comment: This paper commented in: [arXiv:0708.3776], [arXiv:0708.3777],
[arXiv:0708.3779]. Rejoinder in [arXiv:0708.3781]. Published at
http://dx.doi.org/10.1214/088342306000000682 in the Statistical Science
(http://www.imstat.org/sts/) by the Institute of Mathematical Statistics
(http://www.imstat.org
Testing predictor contributions in sufficient dimension reduction
We develop tests of the hypothesis of no effect for selected predictors in
regression, without assuming a model for the conditional distribution of the
response given the predictors. Predictor effects need not be limited to the
mean function and smoothing is not required. The general approach is based on
sufficient dimension reduction, the idea being to replace the predictor vector
with a lower-dimensional version without loss of information on the regression.
Methodology using sliced inverse regression is developed in detail
Rejoinder: Fisher Lecture: Dimension Reduction in Regression
Rejoinder: Fisher Lecture: Dimension Reduction in Regression
[arXiv:0708.3774]Comment: Published at http://dx.doi.org/10.1214/088342307000000078 in the
Statistical Science (http://www.imstat.org/sts/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Algorithms for envelope estimation
Envelopes were recently proposed as methods for reducing estimative variation
in multivariate linear regression. Estimation of an envelope usually involves
optimization over Grassmann manifolds. We propose a fast and widely applicable
one-dimensional (1D) algorithm for estimating an envelope in general. We reveal
an important structural property of envelopes that facilitates our algorithm,
and we prove both Fisher consistency and root-n-consistency of the algorithm.Comment: 30 pages, 2 figures, 2 table
Determining the dimension of iterative Hessian transformation
The central mean subspace (CMS) and iterative Hessian transformation (IHT)
have been introduced recently for dimension reduction when the conditional mean
is of interest. Suppose that X is a vector-valued predictor and Y is a scalar
response. The basic problem is to find a lower-dimensional predictor \eta^TX
such that E(Y|X)=E(Y|\eta^TX). The CMS defines the inferential object for this
problem and IHT provides an estimating procedure. Compared with other methods,
IHT requires fewer assumptions and has been shown to perform well when the
additional assumptions required by those methods fail. In this paper we give an
asymptotic analysis of IHT and provide stepwise asymptotic hypothesis tests to
determine the dimension of the CMS, as estimated by IHT. Here, the original IHT
method has been modified to be invariant under location and scale
transformations. To provide empirical support for our asymptotic results, we
will present a series of simulation studies. These agree well with the theory.
The method is applied to analyze an ozone data set.Comment: Published at http://dx.doi.org/10.1214/009053604000000661 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Principal Fitted Components for Dimension Reduction in Regression
We provide a remedy for two concerns that have dogged the use of principal
components in regression: (i) principal components are computed from the
predictors alone and do not make apparent use of the response, and (ii)
principal components are not invariant or equivariant under full rank linear
transformation of the predictors. The development begins with principal fitted
components [Cook, R. D. (2007). Fisher lecture: Dimension reduction in
regression (with discussion). Statist. Sci. 22 1--26] and uses normal models
for the inverse regression of the predictors on the response to gain reductive
information for the forward regression of interest. This approach includes
methodology for testing hypotheses about the number of components and about
conditional independencies among the predictors.Comment: Published in at http://dx.doi.org/10.1214/08-STS275 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Elevated soil lead: Statistical modeling and apportionment of contributions from lead-based paint and leaded gasoline
While it is widely accepted that lead-based paint and leaded gasoline are
primary sources of elevated concentrations of lead in residential soils,
conclusions regarding their relative contributions are mixed and generally
study specific. We develop a novel nonlinear regression for soil lead
concentrations over time. It is argued that this methodology provides useful
insights into the partitioning of the average soil lead concentration by source
and time over large residential areas. The methodology is used to investigate
soil lead concentrations from the 1987 Minnesota Lead Study and the 1990
National Lead Survey. Potential litigation issues are discussed briefly.Comment: Published at http://dx.doi.org/10.1214/07-AOAS112 in the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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